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\begin{table}[!tbp]
\caption{Simulation results: Exponential outcome model 1\label{tb_exp1}} 
{\centering
\begin{tabular}{lrrcrcrrcrrcrrcrcrr}
\hline
\multicolumn{1}{l}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}\tabularnewline
\cline{2-3} \cline{7-8} \cline{13-14} \cline{18-19}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}\tabularnewline
\hline
{\bfseries Correct PS model}&&&&&&&&&&&&&&&&&&\tabularnewline
~~\textbf{nDBW}&$ -3.99$&$ 11.49$&&$$&&$ -1.08$&$   5.46$&&$$&$$&&$-0.54$&$10.77$&&$$&&$-0.38$&$ 5.01$\tabularnewline
~~MLE&$ -1.28$&$ 29.80$&&$$&&$ -0.10$&$  13.73$&&$$&$$&&$-0.25$&$23.32$&&$$&&$-0.04$&$ 8.80$\tabularnewline
~~CBPS&$  1.51$&$ 13.34$&&$$&&$  0.23$&$   6.47$&&$$&$$&&$ 2.93$&$13.00$&&$$&&$ 0.40$&$ 5.39$\tabularnewline
~~Calibrated weighting&$ -2.67$&$ 11.55$&&$$&&$ -0.75$&$   5.74$&&$$&$$&&$-1.25$&$10.71$&&$$&&$-0.42$&$ 4.84$\tabularnewline
~~Entropy balancing&$ -5.90$&$ 12.08$&&$$&&$ -4.80$&$   6.91$&&$$&$$&&$-5.46$&$11.47$&&$$&&$-4.86$&$ 6.61$\tabularnewline
~~True propensity score&$ -0.11$&$ 49.48$&&$$&&$  0.84$&$  22.47$&&$$&$$&&$-0.77$&$38.44$&&$$&&$-0.47$&$17.41$\tabularnewline
~~Unweighted&$-14.23$&$ 17.95$&&$$&&$-14.34$&$  15.16$&&$$&$$&&$14.79$&$19.57$&&$$&&$14.41$&$15.45$\tabularnewline
~~\textbf{nDBW DR}&$ -3.58$&$ 10.77$&&$$&&$ -0.94$&$   5.06$&&$$&$$&&$-1.10$&$10.49$&&$$&&$-0.32$&$ 4.81$\tabularnewline
~~MLE DR&$ -2.44$&$ 13.31$&&$$&&$ -0.58$&$   6.69$&&$$&$$&&$-0.58$&$12.93$&&$$&&$-0.22$&$ 5.52$\tabularnewline
~~CBPS DR&$ -2.64$&$ 11.91$&&$$&&$ -0.75$&$   5.85$&&$$&$$&&$-0.67$&$11.60$&&$$&&$-0.22$&$ 5.21$\tabularnewline
~~Calibrated weighting DR&$ -2.94$&$ 10.97$&&$$&&$ -0.86$&$   5.31$&&$$&$$&&$-0.88$&$10.69$&&$$&&$-0.28$&$ 4.82$\tabularnewline
~~Entropy balancing DR&$ -5.56$&$ 11.72$&&$$&&$ -4.10$&$   6.35$&&$$&$$&&$-2.60$&$10.61$&&$$&&$-2.05$&$ 4.96$\tabularnewline
~~True propensity score DR~~&$ -2.67$&$ 13.96$&&$$&&$ -0.55$&$   6.98$&&$$&$$&&$-0.85$&$13.00$&&$$&&$-0.31$&$ 5.90$\tabularnewline
~~Imputation&$-10.23$&$ 15.02$&&$$&&$ -9.05$&$  10.40$&&$$&$$&&$-2.38$&$11.67$&&$$&&$-2.55$&$ 5.66$\tabularnewline
\hline
{\bfseries Misspecified PS model}&&&&&&&&&&&&&&&&&&\tabularnewline
~~\textbf{nDBW}&$ -7.91$&$ 13.26$&&$$&&$ -5.49$&$   7.27$&&$$&$$&&$ 4.72$&$12.80$&&$$&&$ 1.24$&$ 5.36$\tabularnewline
~~MLE&$ 57.77$&$470.18$&&$$&&$159.76$&$1503.70$&&$$&$$&&$ 0.43$&$23.32$&&$$&&$ 0.37$&$ 9.51$\tabularnewline
~~CBPS&$  1.78$&$ 12.85$&&$$&&$  0.54$&$   6.41$&&$$&$$&&$10.65$&$18.11$&&$$&&$ 6.21$&$ 8.64$\tabularnewline
~~Calibrated weighting&$ -5.15$&$ 12.06$&&$$&&$ -2.62$&$   6.04$&&$$&$$&&$ 2.61$&$12.17$&&$$&&$ 3.17$&$ 6.08$\tabularnewline
~~Entropy balancing&$ -7.48$&$ 13.07$&&$$&&$ -5.40$&$   7.52$&&$$&$$&&$ 0.83$&$11.72$&&$$&&$ 1.23$&$ 5.15$\tabularnewline
~~True propensity score&$  0.31$&$ 48.20$&&$$&&$ -0.27$&$  22.45$&&$$&$$&&$ 0.38$&$38.69$&&$$&&$-0.28$&$17.85$\tabularnewline
~~Unweighted&$-14.52$&$ 18.28$&&$$&&$-14.28$&$  15.09$&&$$&$$&&$14.28$&$18.98$&&$$&&$14.47$&$15.48$\tabularnewline
~~\textbf{nDBW DR}&$ -5.13$&$ 11.96$&&$$&&$ -2.63$&$   5.88$&&$$&$$&&$ 2.05$&$11.98$&&$$&&$ 2.78$&$ 5.88$\tabularnewline
~~MLE DR&$  0.81$&$ 61.38$&&$$&&$ 45.24$&$ 601.02$&&$$&$$&&$ 3.69$&$14.97$&&$$&&$ 4.11$&$ 7.91$\tabularnewline
~~CBPS DR/BRDR&$ -5.84$&$ 13.06$&&$$&&$ -2.35$&$   6.95$&&$$&$$&&$ 4.23$&$13.75$&&$$&&$ 3.96$&$ 6.94$\tabularnewline
~~Calibrated weighting DR&$ -5.15$&$ 12.06$&&$$&&$ -2.62$&$   6.04$&&$$&$$&&$ 2.61$&$12.17$&&$$&&$ 3.17$&$ 6.08$\tabularnewline
~~Entropy balancing DR&$ -7.48$&$ 13.07$&&$$&&$ -5.40$&$   7.52$&&$$&$$&&$ 0.83$&$11.72$&&$$&&$ 1.23$&$ 5.15$\tabularnewline
~~True propensity score DR~~&$ -2.69$&$ 13.86$&&$$&&$ -0.47$&$   7.26$&&$$&$$&&$-0.99$&$13.08$&&$$&&$-0.34$&$ 5.91$\tabularnewline
~~Imputation&$-10.18$&$ 14.92$&&$$&&$ -9.04$&$  10.32$&&$$&$$&&$-2.47$&$11.95$&&$$&&$-2.50$&$ 5.58$\tabularnewline
\hline
\end{tabular}}
\parbox{0.99\textwidth}
		{Notes: This simulation compares the performance of various methods 
		for estimating propensity scores and (inverse probability) weights 
		by investigating combinations of six versions of the true outcome model 
		(linear~1, linear~2, quadratic~1, quadratic~2, exponential~1, and exponential~2)
		and two versions of coefficients for the true propensity score model (type~A and B)
		with the two different numbers of observations ($n = 200$ and $n = 1000$).
		For each estimation method, I use two propensity score model specifications 
		(correct and misspecified) and report the bias and RMSE for each in the table.}\end{table}
